An 82 foot ladder is leaning against a wall. The is 18 feet from the base of the wall. How high above the ground does the ladder touch the wall?
18^2 + h^2 = 82^2
To find the height above the ground at which the ladder touches the wall, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this problem, the ladder is the hypotenuse, and the distance from the base of the wall to the ladder is one of the other sides. Let's call this distance "x". The height above the ground is the remaining side of the triangle.
Using the Pythagorean theorem, we have:
x^2 + (height)^2 = ladder^2
Substituting the known values:
x^2 + (height)^2 = 82^2
We also know that the distance from the base of the wall to the ladder is 18 feet, so x = 18.
Plugging this into the equation:
18^2 + (height)^2 = 82^2
324 + (height)^2 = 6724
Now, we solve for (height)^2:
(height)^2 = 6724 - 324
(height)^2 = 6400
To find the height, we take the square root of both sides:
height = √6400
height ≈ 80 feet
Therefore, the ladder touches the wall at a height of approximately 80 feet above the ground.